arXiv subjects

Mikhail V. Bondarko

Publications and source records attributed to Mikhail V. Bondarko.

At least 37 records · Page 2Linked to original sources

arXiv2019

On purely generated $α$-smashing weight structures and weight-exact localizations

This paper is dedicated to new methods of constructing weight structures and weight-exact localizations; our arguments generalize their bounded versions considered in previous papers of the authors. We start from a class of objects $P$ of triangulated category $C$ that satisfies a certain negativity condition (there are no $C$-extensions of positive degrees between elements of $P$; we actually need a somewhat stronger condition of this sort) to obtain a weight structure both "halves" of which are closed either with respect to $C$-coproducts of less than $α$ objects (for $α$ being a fixed regular cardinal) or with respect to all coproducts (provided that $C$ is closed with respect to coproducts of this sort). This construction gives all "reasonable" weight structures satisfying the latter condition. In particular, we obtain certain weight structures on spectra (in $SH$) consisting of less than $α$ cells and on certain localizations of $SH$; these results are new.

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arXiv2018

Conservativity of realizations implies that numerical motives are Kimura-finite and motivic zeta functions are rational

We prove: if the (étale or de Rham) realization functor is conservative on the category $DM_{gm}$ of Voevodsky motives with rational coefficients then motivic zeta functions of arbitrary varieties are rational and numerical motives are Kimura-finite. The latter statement immediately implies that the category of numerical motives is (essentially) Tannakian. This observation becomes actual due to the recent announcement of J. Ayoub that the De Rham cohomology realization is conservative on $DM_{gm}(k)$ whenever $\operatorname{char} k=0$. We apply this statement to exterior powers of motives coming from generic hyperplane sections of smooth affine varieties.

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arXiv2018

On Chow weight structures without projectivity and resolution of singularities

In this paper certain Chow weight structures on the "big" triangulated motivic categories $DM_R^{eff}\subset DM_R$ are defined in terms of motives of all smooth varieties over the base field. This definition allows studying basic properties of these weight structures without applying resolution of singularities; thus we don't have to assume that the coefficient ring $R$ contains $1/p$ in the case where the characteristic $p$ of the base field is positive. Moreover, in the case where $R$ satisfies the latter assumption our weight structures are "compatible" with Chow weight structures defined in previous papers (in terms of Chow motives). The results of this article yield certain Chow-weight filtration (also) on $p$-adic cohomology of motives and smooth varieties.

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arXiv2018

Gersten weight structures for motivic homotopy categories; retracts of cohomology of function fields, motivic dimensions, and coniveau spectral sequences

For any cohomology theory $H$ that can be factorized through (the Morel-Voevodsky's triangulated motivic homotopy category) $SH^{S^1}(k)$ we establish the $SH^{S^1}(k)$-functoriality of coniveau spectral sequences for $H$. We also prove: for any affine essentially smooth semi-local $S$ the Cousin complex for $H^*(S)$ splits; if $H$ also factorizes through $SH^+(k)$ or $DM(k)$, then this is also true for any primitive $S$. Moreover, the cohomology of such an $S$ is a direct summand of the cohomology of any its open dense subscheme. This is a vast generalization of the results of a previous paper. In order to prove these results we consider certain triangulated categories of motivic pro-spectra, and introduce Gersten weight structures for them. We study in detail the notions of cohomological dimensions of scheme associated to various categories of motivic pro-spectra; they are defined in terms of the corresponding weight structures and count the number of non-zero Nisnevich cohomology for sheaves in the hearts of orthogonal "homotopy" $t$-structures.

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arXiv2017

On relative $K$-motives, weights for them, and negative $K$-groups

We study certain triangulated categories of $K$-motives $DK(-)$ over a wide class of base schemes, and define certain "weights" for them. We relate the weights of particular $K$-motives to (negative) homotopy invariant $K$-groups (tensored by $\mathbb{Z}[S^{-1}]$ for $S$ being the set of "non-invertible primes") $\mathcal{K}_*(-)$. Our results yield a (new) result on the vanishing of $\mathcal{K}_i(-)$ and of relative $\mathcal{K}$-groups for $i$ being "too negative"; this statement is closely related to a question of Ch. Weibel. We also prove that $\mathcal{K}_i(-)$ for $i<0$ is "supported in codimension $-i$". Moreover, we establish several criteria for bounding (below) the weights of $f_*(1_Y)$ ($1_Y$ is the tensor unit in $DK(Y)$); this automatically implies the vanishing of the corresponding $E_2$-terms of Chow-weight spectral sequences (and of the factors of the corresponding Chow-weight filtrations) for any (co)homology of these motives. Our methods of bounding weights can be applied to various "motivic" triangulated categories; this yields some new statements on (constructible) complexes of étale sheaves. We also relate the weights of $K$-motives with rational coefficients to that of Beilinson motives; the Chow-weight spectral sequences converging to their $\mathbb{Q}_l$-étale (co)homology yield Deligne-type weights for the latter. Somewhat surprisingly, we are able to prove in certain ("extreme") cases that the corresponding weight bounds coming from étale (co)homology are precise. We illustrate these statements by simple examples.

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arXiv2017

On morphisms killing weights, weight complexes, and Eilenberg-Maclane (co)homology of spectra

We ask whether a morphism $g$ in a triangulated category $C$ endowed with a weight structure "kills weights" (between an integer $m$ and some $n\ge m$). If $g=id_M$ (where $M\in Obj C$) and $C$ is Karoubian, then $g$ kills weights $m,\dots,n$ whenever there exists a weight decomposition of $M$ that "avoids" these weights (in the sense earlier defined by Wildeshaus). We prove the equivalence of several definitions for killing weights. In particular, we describe a family of cohomological functors that "detects" this notion. We also prove that $M$ is without weights $m,\dots, n$ (i.e., a decomposition of $M$ avoiding these weights exists) if and only if the corresponding condition is fulfilled for its weight complex $t(M)$. These results allow us to get new (stronger) results on the conservativity of the weight complex functor $t$. We study in detail the case $C=SH$ (endowed with the spherical weight structure whose heart consists of coproducts of sphere spectra); the corresponding weight complex functor is just the one calculating the $H\mathbb{Z}$-homology (whose terms are free abelian groups). In this case $g$ kills weights $m,\dots, n$ if and only if $H(g)=0$ for all $H$ represented by elements of $SH[m,n]$ (so, these morphisms form an injective class of morphisms in the sense defined by Christensen; yet this class is not stable with respect to shifts). Moreover, for any spectrum $M$ there exists a "weakly universal decomposition" $P\to M\to I_0$ for $I_0\in SH[m,n]$ and $P$ being without weights $m,\dots,n$ (so, we obtain a torsion pair). We also prove a certain converse to the stable Hurewicz theorem.

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arXiv2017

Intersecting the dimension filtration with the slice one for (relative) motivic categories

In this paper we prove that the intersections of the levels of the dimension filtration on Voevodsky's motivic complexes over a field $k$ with the levels of the slice one are "as small as possible", i.e., that $Obj d_{\le m}DM^{eff}_{-,R} \cap Obj DM^{eff}_{-,R} (i)=Obj d_{\le m-i} DM^{eff}_{-,R} (i)$ (for $m,i\ge 0$ and $R$ being any coefficient ring in which the exponential characteristic of $k$ invertible). This statement is applied to prove that a conjecture of J. Ayoub is equivalent to a certain orthogonality assumption. We also establish a vast generalization of our intersection result to relative motivic categories (that are required to fulfil a certain list of "axioms"). In the process we prove several new properties of relative motives and of the so-called Chow weight structures for them.

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arXiv2017

On constructing weight structures and extending them to idempotent extensions

We describe a new method for constructing a weight structure $w$ on a triangulated category $C$. For a given $C$ and $w$ it allow us to give a fairly comprehensive (and new) description of those triangulated categories consisting of retracts of objects of $C$ (i.e., of subcategories of the Karoubi envelope of $C$ that contain $C$; we call them idempotent extensions of $C$) such that $w$ extends to them. In particular, any bounded above or below $w$ extends to any idempotent extension of $C$. We also discuss the applications of our results to certain triangulated categories of ("relative") motives.

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arXiv2016

A Nullstellensatz for triangulated categories

The main goal of this paper is to prove the following: for a triangulated category $ \underline{C}$ and $E\subset \operatorname{Obj} \underline{C}$ there exists a cohomological functor $F$ (with values in some abelian category) such that $E$ is its set of zeros if (and only if) $E$ is closed with respect to retracts and extensions (so, we obtain a certain Nullstellensatz for functors of this type). Moreover, for $ \underline{C}$ being an $R$-linear category (where $R$ is a commutative ring) this is also equivalent to the existence of an $R$-linear $F: \underline{C}^{op}\to R-\operatorname{mod}$ satisfying this property. As a corollary, we prove that an object $Y$ belongs to the corresponding "envelope" of some $D\subset \operatorname{Obj} \underline{C}$ whenever the same is true for the images of $Y$ and $D$ in all the categories $ \underline{C}_p$ obtained from $ \underline{C}$ by means of "localizing the coefficients" at maximal ideals $p\triangleleft R$. Moreover, to prove our theorem we develop certain new methods for relating triangulated categories to their (non-full) countable triangulated subcategories. The results of this paper can be applied to the study of weight structures and of triangulated categories of motives.

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arXiv2015

On Chow weight structures for $cdh$-motives with integral coefficients

The main goal of this paper is to define a certain Chow weight structure $w_{Chow}$ on the category $DM_c(S)$ of (constructible) $cdh$-motives over an equicharacteristic scheme $S$. In contrast to the previous papers of D. Hébert and the first author on weights for relative motives (with rational coefficients), we can achieve our goal for motives with integral coefficients (if $\operatorname{char}S=0$; if $\operatorname{char}S=p>0$ then we consider motives with $\mathbb{Z}[\frac{1}{p}]$-coefficients). We prove that the properties of the Chow weight structures that were previously established for $\mathbb{Q}$-linear motives can be carried over to this "integral" context (and we generalize some of them using certain new methods). In this paper we mostly study the version of $w_{Chow}$ defined via "gluing from strata"; this enables us to define Chow weight structures for a wide class of base schemes. As a consequence, we certainly obtain certain (Chow)-weight spectral sequences and filtrations for any (co)homology of motives.

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arXiv2015

Gersten weight structures for motivic homotopy categories; direct summands of cohomology of function fields and coniveau spectral sequences

For any cohomology theory $H$ that can be factorized through (the Morel-Voevodsky's triangulated motivic homotopy category) $SH^{S^1}(k)$ (or through $SH(k)$) we establish the $SH^{S^1}(k)$-functorialty (resp. $SH(k)$-one) of coniveau spectral sequences for $H$. We also prove: for any affine essentially smooth semi-local $S$ the Cousin complex for $H^*(S)$ splits; if $H$ also factorizes through $SH^+(k)$ or $SH^{MGL}(k)$, then this is also true for any primitive $S$. Moreover, the cohomology of such an $S$ is a direct summand of the cohomology of any its open dense subscheme. This is a vast generalization of the results of a previous paper. In order to prove these results we consider certain categories of motivic pro-spectra, and introduce Gersten weight structures for them. Our results rely on several interesting statements on weight structures in cocompactly cogenerated triangulated categories and on the '$SH^+(k)$-acyclity' of primitive schemes. .

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arXiv2015

On perverse homotopy $t$-structures, coniveau spectral sequences, cycle modules, and relative Gersten weight structures

We study the category $DM(S)$ of Beilinson motives (as described by Cisinski and Deglise) over a more or less general base scheme $S$, and establish several nice properties for a version $t_{hom}(S)$ of the perverse homotopy $t$-structure (essentially defined by Ayoub) for it. $t_{hom}(S)$ is characterized in terms of certain stalks of an $S$-motif $H$ and its Tate twists at fields over $S$; it is closely related to certain coniveau spectral sequences for the cohomology of (the Borel-Moore motives of) arbitrary finite type $S$-schemes. We conjecture that the heart of $t_{hom}(S)$ is given by cycle modules over $S$ (as defined by Rost); for varieties over characteristic $0$ fields this conjecture was recently proved by Deglise. Our definition of $t_{hom}(S)$ is closely related to a new effectivity filtration for $DM(S)$ (and for the subcategory of Chow $S$-motives in it). We also sketch the construction of a certain Gersten weight structure for the category of $S$-comotives; this weight structure yields one more description of $t_{hom}(S)$ and its heart.

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arXiv2015

Weights for relative motives; relation with mixed complexes of sheaves

The main goal of this paper is to define the so-called Chow weight structure for the category of Beilinson motives over any 'reasonable' base scheme $S$ (this is the version of Voevodsky's motives over $S$ defined by Cisinski and Deglise). We also study the functoriality properties of the Chow weight structure (they are very similar to the well-known functoriality of weights for mixed complexes of sheaves). As shown in a preceding paper, the Chow weight structure automatically yields an exact conservative weight complex functor (with values in $K^b(Chow(S))$). Here $Chow(S)$ is the heart of the Chow weight structure; it is 'generated' by motives of regular schemes that are projective over $S$. Besides, Grothendiek's group of $S$-motives is isomorphic to $K_0(Chow(S))$; we also define a certain 'motivic Euler characteristic' for $S$-schemes. We obtain (Chow)-weight spectral sequences and filtrations for any cohomology of motives; we discuss their relation to Beilinson's 'integral part' of motivic cohomology and to weights of mixed complexes of sheaves. For the study of the latter we introduce a new formalism of relative weight structures.

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arXiv2015

A 'fat hyperplane section' weak Lefschetz (in arbitrary characteristic), and Barth-type theorems

We prove a certain 'fat hyperplane section' Weak Lefschetz-type theorem for etale cohomology of non-projective varieties, similar to a result of Goresky and MacPherson (over complex numbers). This statement easily yields certain (vast) generalizations of the 'ordinary' Weak Lefschetz and Barth's theorems in arbitrary characteristic (that do not require any stratified Morse theory for their proof).

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arXiv2014

Non-commutative localizations of additive categories and weight structures; applications to birational motives

In this paper we demonstrate that 'non-commutative localizations' of arbitrary additive categories (generalizing those defined by Cohn for rings) are closely (and naturally) related with weight structures. Localizing an arbitrary triangulated $C$ by a set $S$ of morphisms in the heart of a weight structure $w$ for it one obtains a triangulated category endowed with a weight structure $w'$. The heart of $w'$ is a certain idempotent completion of the non-commutative localization of the heart of $w$ by $S$. The latter is the natural categorical version of Cohn's localizations of rings i.e. the functor connecting hearts is universal among all the additive functors that make the elements of $S$ invertible. In particular, taking $C=K^b(A)$ for an additive $A$ we obtain a very efficient tool for computing the additive localization of $A$ by $S$; using it, we generalize the calculations of Gerasimov and Malcolmson. We apply our results to certain categories of birational motives over a base scheme $U$ (generalizing those defined by Kahn and Sujatha). When $U$ is the spectrum of a perfect field, the weight structure obtained is compatible with the Chow and Gersten weight structures defined by the first author in previous papers. For a general $U$ the result is completely new. We also consider the relation of weight structures with their adjacent t-structures (in localizations). In the 'motivic' setting mentioned this yields the natural generalization of the 'duality' between birational motives and birational sheaves with transfers established by Kahn and Sujatha.

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arXiv2014

Mixed motivic sheaves (and weights for them) exist if 'ordinary' mixed motives do

The goal of this paper is to prove: if certain 'standard' conjectures on motives over algebraically closed fields hold, then over any 'reasonable' $S$ there exists a motivic $t$-structure for the category of Voevodsky's $S$-motives (as constructed by Cisinski and Deglise). If $S$ is 'very reasonable' (for example, of finite type over a field) then the heart of this $t$-structure (the category of mixed motivic sheaves over $S$) is endowed with a weight filtration with semi-simple factors. We also prove a certain 'motivic decomposition theorem' (assuming the conjectures mentioned) and characterize semi-simple motivic sheaves over $S$ in terms of those over its residue fields. Our main tool is the theory of weight structures. We actually prove somewhat more than the existence of a weight filtration for mixed motivic sheaves: we prove that the motivic $t$-structure is transversal to the Chow weight structure for $S$-motives (that was introduced previously and independently by D. Hebert and the author; weight structures and their transversality with t-structures were also defined by the author in recent papers). We also deduce several properties of mixed motivic sheaves from this fact. Our reasoning relies on the degeneration of Chow-weight spectral sequences for 'perverse 'etale homology' (that we prove unconditionally); this statement also yields the existence of the Chow-weight filtration for such (co)homology that is strictly restricted by ('motivic') morphisms.

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arXiv2013

On weights for relative motives with integral coefficients

The goal of this paper is to define a certain Chow weight structure for the category of Voevodsky's motivic complexes with integral coefficients (as described by Cisinski and Deglise) over any excellent finite-dimensional separated scheme $S$. Our results are parallel to (though substantially weaker than) the corresponding 'rational coefficient' statements proved by D. Hebert and the author. As an immediate consequence of the existence of 'weights', we obtain certain (Chow)-weight spectral sequences and filtrations for any (co)homology of $S$-motives.

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arXiv2011

Weights and t-structures: in general triangulated categories, for 1-motives, mixed motives, and for mixed Hodge complexes and modules

We study certain 'weights' for triangulated categories endowed with $t$-structures. Our results axiomatize and describe in detail the relations between the Chow weight structure (introduced in a preceding paper), the (conjectural) motivic t-structure for Voevodsky's motives, and the conjectural weight filtration for them. This picture becomes non-conjectural when restricted to the derived categories of 1-motives (over a smooth base) and of Artin-Tate motives over number fields; other examples include Beilinson's derived category of graded polarizable mixed Hodge complexes and Saito's derived category of mixed Hodge modules. We also study weight filtrations for the heart of $t$ and (the degeneration of) weight spectral sequences. In a subsequent paper we apply the results obtained in order to prove that (certain) 'standard' conjectures imply the existence of Beilinson's categories of mixed motivic sheaves endowed with weight filtrations.

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